# Direction of static friction in rolling without slipping down an inclined plane

Suppose a wheel is rolling without slipping down an incline. Static friction prevents the impending motion of the wheel relative to the incline. Since the slip is impending in the direction in which the wheel rolls, shouldn't the static friction point opposite to this direction or downwards along the incline?

• Your diagram needs explanation. What is τ in your diagram? Is it some applied external torque to the wheel? The problem statement makes it sound like the only force acting on the wheel is gravity. Now it is confusing as to what the actual forces are acting on the wheel. Please clarify so we can answer appropriately. May 27, 2020 at 21:11
• I think this may help you. May 28, 2020 at 1:35

First, think about how the surfaces would slip without friction. In this case the wheel would slide down the incline without rolling. Static friction will therefore try to prevent this, and so must point up the incline.

Another way to think of it: you have assumed rolling without slipping. The only force that exerts a torque about the center of mass of the wheel is static friction, so this force needs to be responsible in causing the rotation of the wheel to match up with the linear motion so that slipping doesn't occur. If friction pointed down the incline, we would get slipping because the wheel cannot rotate according to that torque and move without slipping. Referring to your image, movement down the incline needs to be matched with clockwise rotation to have rolling without slipping.

It looks like you are considering a scenario where some other force tries to spin the wheel in the clockwise direction, but this involves another force acting on the wheel that has a torque about the center of mass of the wheel. In that case then the analysis becomes different, and the direction of static friction can depend on the magnitude and location of the force, as well as the moment of inertia of the object.

As a small note, I wouldn't put both the force and torque of friction on your free body diagram. Usually a torque on a free body diagram indicates in reality two forces that are equal but opposite that have a zero net force but a non-zero net torque. Therefore, I would just draw the friction force, not its torque as well.

• Thank you for the explanation. The reason I thought the static friction should point upwards is that if it were absent then gravity causes downwards or clockwise rotation meaning slip is impending clockwise. So I thought static friction must point counterclockwise to counteract the impending slip. What's wrong with this reasoning? May 28, 2020 at 21:35
• @user436788 Gravity doesn't cause any rotation. Without friction the wheel would not rotate at all down the incline. It would just slide. May 28, 2020 at 21:38
• Ok so because only forces that are not located at the mass center cause torques relative to the mass center these are the only forces that cause rotation. Thank you. May 28, 2020 at 22:24

Slip is not "impending in the direction in which the wheel rolls". Slip is impending in the opposite direction to which which the present forces are pushing it.

And the force to consider here is gravity; it is trying to make the contact point slip by pulling the ball downwards, so static friction must pull upwards to prevent the contact point from slipping.

A fruitful way of thinking about this is by imaging a star rolling down the inlince. (On this image the star is on flat ground, but imagine the ground being tilted.)

• To have "rolling" without slipping, each leg must not slip while in contact with the incline surface. Gravity pulls downwards, so static friction must pull upwards to avoid that the leg slides down.

• With more legs, the same is still the case. Each leg takes over right as the previous leg let's go, but while it is in contact it mustn't slide. Gravity causes this sliding, so static friction must point upwards.

• With even more legs, the same is still the case.

• With so many legs that we basically have a continuous circular surface - with infinitely many legs that are infinitely close but also infinitely small, so just one point each. Each point is still a leg, so for this wheel, the above description still counts while a point is touching the surface: Gravity pulls down trying to make it slide, so static friction must pull up to prevent it.